Kiefer C. Quantum gravity

PROBLEMS OF A FUNDAMENTALLY SEMICLASSICAL THEORY 19
between the metric corresponding to a superposition A|Ψ 1 〉 + B|Ψ 2 〉 (which still
satisfies the Schrödinger equation) and the metrics g 1 and g 2 . This was already
remarked by Anderson in Møller (1962) and by Belinfante in a discussion with
Rosenfeld (see Infeld 1964). According to von Borzeszkowski and Treder (1988)
it was also the reason why Dirac strongly objected to (1.35).
Rosenfeld insisted on (1.35) because he strongly followed Bohr’s interpretation
of the measurement process for which classical concepts should be indispensable.
This holds in particular for the structure of space–time, so he wished
to have a c-number representation for the metric. He rejected a quantum description
for the total system and answered to Belinfante in Infeld (1964) that
Einstein’s equations may merely be thermodynamical equations of state that
break down for large fluctuations, that is, the gravitational field may only be an
effective, not a fundamental, field; cf. also Jacobson (1995).
The problem with the superposition principle can be demonstrated by the
following argument that has even been put to an experimental test (Page and
Geilker 1981). One assumes that there is no explicit collapse of |Ψ〉, because
otherwise one would expect the covariant conservation law 〈 ˆT µν 〉; ν =0tobe
violated, in contradiction to (1.35). If the gravitational field were quantized, one
would expect that each component of the superposition in |Ψ〉 would act as a
source for the gravitational field. This is of course the Everett interpretation
of quantum theory; cf. Chapter 10. On the other hand, the semiclassical Einstein
equations (1.35) depend on all components of |Ψ〉 simultaneously. Page
and Geilker (1981) envisaged the following gedanken experiment, reminiscent of
Schrödinger’s cat, to distinguish between these options.
In a box, there is a radioactive source together with two masses that are
connected by a spring. Initially, the masses are rigidly connected, so that they
cannot move. If a radioactive decay happens, the rigid connection will be broken
and the masses can swing towards each other. Outside the box, there is a
Cavendish balance that is sensitive to the location of the masses and therefore
acts as a device to ‘measure’ their position. Following Unruh (1984), the situation
can be described by the following simple model. We denote with |0〉, the
quantum state of the masses with rigid connection, and with |1〉, the corresponding
state in which they can move towards each other. For the purpose of this
experiment, it is sufficient to go to the Newtonian approximation of GR and to
use the Hamilton operator Ĥ instead of the full energy–momentum tensor ˆT µν .
For initial time t = 0, it is assumed that the state is given by |0〉. Fort>0, the
state then evolves into a superposition of |0〉 and |1〉,
|Ψ〉(t) =α(t)|0〉 + β(t)|1〉 ,
with the coefficients |α(t)| 2 ≈ e −λt , |β| 2 ≈ 1 − e −λt , according to the law of
radioactive decay, with a decay constant λ. From this, one finds for the evolution
of the expectation value
[ ]
〈Ψ|Ĥ|Ψ〉(t) =|α(t)|2 〈0|Ĥ|0〉 + |β(t)|2 〈1|Ĥ|1〉 +2Re α ∗ β〈0|Ĥ|1〉 .